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We have already seen (12.15.11) that (a) implies (b). If u is weakly con-
tinuous, then for all y e E the linear form x\-+(u(x) \y) is weakly continuous,
and therefore a fortiori continuous with respect to the strong topology, which
is finer than the weak topology. Consequently (63.2), for each y e E there
exists a unique point u*(y) such that (u(x)\y) = (x\u*(yj) for all xe E: in
other words (11.5), u has an adjoint, and therefore (b) implies (c). Finally, if u
has an adjoint, then we have

Htt*GOII = sup |(*|ii*(y))|= sup \(u(x)\y)\,

\\x\\l                              ||*||SS1

and each of the linear forms y*-+(y\u(x)) = (u(x)\y) is continuous, hence
y\~+ \\u*(y)\\ is a lower semicontinuous seminorm on E, and is therefore con-
tinuous in the strong topology (12.16.3). But this is equivalent to saying that
u* is (strongly) continuous, and therefore so is u = ** (11.5.2).

We remark that it is essential for the truth of (12.16.4), (12.16.5), (12.16.6),
and (12.16.7) that the space E should be complete (cf. Problems 21 and 22).

(12.16.8) (Banach's theorem) Let E, F be two Frechet spaces and let ubea
continuous linear mapping ofE into F. Then either u(E) is meager in F, or else
u(E) = F. In the latter case, if N is the kernel of u and E - E/N -^ F the
canonical factorization of u, the mapping v is an isomorphism of the Frtchet
space E/N (12.14.9) onto F (in other words (12.12.7), u is a strict morphism of
E onto F).

Suppose that u(E) is not meager in F. By virtue of (12.13.1) and (12.12.7), the
theorem will be proved if we can show that, for each neighborhood V of 0 in
E, w(V) is a neighborhood of 0 in F. There are two steps to the proof.

( Let E, F be two topological vector spaces, and u a linear mapping
ofE into F such that u(E) is not meager in F. Then for each neighborhood V 0/0
in E, the closure w(V) of the image ofV is a neighborhood 0/0 in F.

Let W be a balanced neighborhood of 0 in E such that W + W <= V
((12.13.1) and (12.8.3)). Then for each *eE there exists an integer n^ 1
such that x e W. Consequently u(E) is the union of the sets w(wW) = n - w(W).
Since u(E) is not meager, at least one of the sets n  w(W) has an interior
point, and therefore so does #(W). But since -u(W) = w(W), we have
~w(W) = w(W). So if y0 is an interior point of w(W), then so is -j>0, and
therefore 0=y0 + (-J0) is an interior point of w(W) 4- w(W) (12.8.2). But
w(W) + w(W) is contained in the closure of w(W) + w(W) = w(W + W)c w(V),
and so ( is proved.ubspaces of E. Deduce that T(J? - x) contains only one -^-invariant point.